Module mathcomp.analysis.probability
From mathcomp Require Import all_ssreflect.From mathcomp Require Import ssralg poly ssrnum ssrint interval finmap.
From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
From mathcomp Require Import cardinality fsbigop.
From HB Require Import structures.
Require Import exp numfun lebesgue_measure lebesgue_integral.
Require Import reals ereal signed topology normedtype sequences esum measure.
Require Import exp numfun lebesgue_measure lebesgue_integral kernel.
# Probability
This file provides basic notions of probability theory. See measure.v for
the type probability T R (a measure that sums to 1).
```
{RV P >-> R} == real random variable: a measurable function from
the measurableType of the probability P to R
distribution P X == measure image of the probability measure P by the
random variable X : {RV P -> R}
P as type probability T R with T of type
measurableType.
Declared as an instance of probability measure.
'E_P[X] == expectation of the real measurable function X
covariance X Y == covariance between real random variable X and Y
'V_P[X] == variance of the real random variable X
mmt_gen_fun X == moment generating function of the random variable
X
{dmfun T >-> R} == type of discrete real-valued measurable functions
{dRV P >-> R} == real-valued discrete random variable
dRV_dom X == domain of the discrete random variable X
dRV_enum X == bijection between the domain and the range of X
pmf X r := fine (P (X @^-1` [set r]))
enum_prob X k == probability of the kth value in the range of X
```
```
bernoulli_pmf p == Bernoulli pmf
bernoulli p == Bernoulli probability measure when 0 <= p <= 1
and \d_false otherwise
binomial_pmf n p == binomial pmf
binomial_prob n p == binomial probability measure when 0 <= p <= 1
and \d_0%N otherwise
bin_prob n k p == $\binom{n}{k}p^k (1-p)^(n-k)$
Computes a binomial distribution term for
k successes in n trials with success rate p
uniform_pdf a b == uniform pdf
uniform_prob a b ab == uniform probability over the interval [a,b]
with ab0 a proof that 0 < b - a
```
Reserved Notation "'{' 'RV' P >-> R '}'"
(at level 0, format "'{' 'RV' P '>->' R '}'").
Reserved Notation "''E_' P [ X ]" (format "''E_' P [ X ]", at level 5).
Reserved Notation "''V_' P [ X ]" (format "''V_' P [ X ]", at level 5).
Reserved Notation "{ 'dmfun' aT >-> T }"
(at level 0, format "{ 'dmfun' aT >-> T }").
Reserved Notation "'{' 'dRV' P >-> R '}'"
(at level 0, format "'{' 'dRV' P '>->' R '}'").
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory GRing.Theory Num.Def Num.Theory.
Import numFieldTopology.Exports.
Local Open Scope classical_set_scope.
Local Open Scope ring_scope.
Definition random_variable (d : _) (T : measurableType d) (R : realType)
(P : probability T R) := {mfun T >-> R}.
Notation "{ 'RV' P >-> R }" := (@random_variable _ _ R P) : form_scope.
Lemma notin_range_measure d (T : measurableType d) (R : realType)
(P : {measure set T -> \bar R}) (X : T -> R) r :
r \notin range X -> P (X @^-1` [set r]) = 0%E.
Proof.
Lemma probability_range d (T : measurableType d) (R : realType)
(P : probability T R) (X : {RV P >-> R}) : P (X @^-1` range X) = 1%E.
Proof.
Definition distribution d (T : measurableType d) (R : realType)
(P : probability T R) (X : {mfun T >-> R}) :=
pushforward P (@measurable_funP _ _ _ X).
Section distribution_is_probability.
Context d (T : measurableType d) (R : realType) (P : probability T R)
(X : {mfun T >-> R}).
Let distribution0 : distribution P X set0 = 0%E.
Proof.
Let distribution_ge0 A : (0 <= distribution P X A)%E.
Proof.
Let distribution_sigma_additive : semi_sigma_additive (distribution P X).
Proof.
HB.instance Definition _ := isMeasure.Build _ _ R (distribution P X)
distribution0 distribution_ge0 distribution_sigma_additive.
Let distribution_is_probability : distribution P X [set: _] = 1%:E.
Proof.
HB.instance Definition _ := Measure_isProbability.Build _ _ R
(distribution P X) distribution_is_probability.
End distribution_is_probability.
Section transfer_probability.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Lemma probability_distribution (X : {RV P >-> R}) r :
P [set x | X x = r] = distribution P X [set r].
Proof.
by []. Qed.
Lemma integral_distribution (X : {RV P >-> R}) (f : R -> \bar R) :
measurable_fun [set: R] f -> (forall y, 0 <= f y) ->
\int[distribution P X]_y f y = \int[P]_x (f \o X) x.
Proof.
End transfer_probability.
HB.lock Definition expectation {d} {T : measurableType d} {R : realType}
(P : probability T R) (X : T -> R) := (\int[P]_w (X w)%:E)%E.
Canonical expectation_unlockable := Unlockable expectation.unlock.
Arguments expectation {d T R} P _%R.
Notation "''E_' P [ X ]" := (@expectation _ _ _ P X) : ereal_scope.
Section expectation_lemmas.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Lemma expectation_fin_num (X : {RV P >-> R}) : P.-integrable setT (EFin \o X) ->
'E_P[X] \is a fin_num.
Proof.
Lemma expectation_cst r : 'E_P[cst r] = r%:E.
Proof.
Lemma expectation_indic (A : set T) (mA : measurable A) : 'E_P[\1_A] = P A.
Proof.
Lemma integrable_expectation (X : {RV P >-> R})
(iX : P.-integrable [set: T] (EFin \o X)) : `| 'E_P[X] | < +oo.
Proof.
move: iX => /integrableP[? Xoo]; rewrite (le_lt_trans _ Xoo)// unlock.
exact: le_trans (le_abse_integral _ _ _).
Qed.
exact: le_trans (le_abse_integral _ _ _).
Qed.
Lemma expectationM (X : {RV P >-> R}) (iX : P.-integrable [set: T] (EFin \o X))
(k : R) : 'E_P[k \o* X] = k%:E * 'E_P [X].
Proof.
Lemma expectation_ge0 (X : {RV P >-> R}) :
(forall x, 0 <= X x)%R -> 0 <= 'E_P[X].
Proof.
Lemma expectation_le (X Y : T -> R) :
measurable_fun [set: T] X -> measurable_fun [set: T] Y ->
(forall x, 0 <= X x)%R -> (forall x, 0 <= Y x)%R ->
{ae P, (forall x, X x <= Y x)%R} -> 'E_P[X] <= 'E_P[Y].
Proof.
move=> mX mY X0 Y0 XY; rewrite unlock ae_ge0_le_integral => //.
- by move=> t _; apply: X0.
- exact/EFin_measurable_fun.
- by move=> t _; apply: Y0.
- exact/EFin_measurable_fun.
- move: XY => [N [mN PN XYN]]; exists N; split => // t /= h.
by apply: XYN => /=; apply: contra_not h; rewrite lee_fin.
Qed.
- by move=> t _; apply: X0.
- exact/EFin_measurable_fun.
- by move=> t _; apply: Y0.
- exact/EFin_measurable_fun.
- move: XY => [N [mN PN XYN]]; exists N; split => // t /= h.
by apply: XYN => /=; apply: contra_not h; rewrite lee_fin.
Qed.
Lemma expectationD (X Y : {RV P >-> R}) :
P.-integrable [set: T] (EFin \o X) -> P.-integrable [set: T] (EFin \o Y) ->
'E_P[X \+ Y] = 'E_P[X] + 'E_P[Y].
Proof.
Lemma expectationB (X Y : {RV P >-> R}) :
P.-integrable [set: T] (EFin \o X) -> P.-integrable [set: T] (EFin \o Y) ->
'E_P[X \- Y] = 'E_P[X] - 'E_P[Y].
Proof.
Lemma expectation_sum (X : seq {RV P >-> R}) :
(forall Xi, Xi \in X -> P.-integrable [set: T] (EFin \o Xi)) ->
'E_P[\sum_(Xi <- X) Xi] = \sum_(Xi <- X) 'E_P[Xi].
Proof.
elim: X => [|X0 X IHX] intX; first by rewrite !big_nil expectation_cst.
have intX0 : P.-integrable [set: T] (EFin \o X0).
by apply: intX; rewrite in_cons eqxx.
have {}intX Xi : Xi \in X -> P.-integrable [set: T] (EFin \o Xi).
by move=> XiX; apply: intX; rewrite in_cons XiX orbT.
rewrite !big_cons expectationD ?IHX// (_ : _ \o _ = fun x =>
\sum_(f <- map (fun x : {RV P >-> R} => EFin \o x) X) f x).
by apply: integrable_sum => // _ /mapP[h hX ->]; exact: intX.
by apply/funext => t/=; rewrite big_map sumEFin mfun_sum.
Qed.
have intX0 : P.-integrable [set: T] (EFin \o X0).
by apply: intX; rewrite in_cons eqxx.
have {}intX Xi : Xi \in X -> P.-integrable [set: T] (EFin \o Xi).
by move=> XiX; apply: intX; rewrite in_cons XiX orbT.
rewrite !big_cons expectationD ?IHX// (_ : _ \o _ = fun x =>
\sum_(f <- map (fun x : {RV P >-> R} => EFin \o x) X) f x).
by apply: integrable_sum => // _ /mapP[h hX ->]; exact: intX.
by apply/funext => t/=; rewrite big_map sumEFin mfun_sum.
Qed.
End expectation_lemmas.
HB.lock Definition covariance {d} {T : measurableType d} {R : realType}
(P : probability T R) (X Y : T -> R) :=
'E_P[(X \- cst (fine 'E_P[X])) * (Y \- cst (fine 'E_P[Y]))]%E.
Canonical covariance_unlockable := Unlockable covariance.unlock.
Arguments covariance {d T R} P _%R _%R.
Section covariance_lemmas.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Lemma covarianceE (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) ->
P.-integrable setT (EFin \o Y) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
covariance P X Y = 'E_P[X * Y] - 'E_P[X] * 'E_P[Y].
Proof.
move=> X1 Y1 XY1.
have ? : 'E_P[X] \is a fin_num by rewrite fin_num_abs// integrable_expectation.
have ? : 'E_P[Y] \is a fin_num by rewrite fin_num_abs// integrable_expectation.
rewrite unlock [X in 'E_P[X]](_ : _ = (X \* Y \- fine 'E_P[X] \o* Y
\- fine 'E_P[Y] \o* X \+ fine ('E_P[X] * 'E_P[Y]) \o* cst 1)%R); last first.
apply/funeqP => x /=; rewrite mulrDr !mulrDl/= mul1r fineM// mulrNN addrA.
by rewrite mulrN mulNr [Z in (X x * Y x - Z)%R]mulrC.
have ? : P.-integrable [set: T] (EFin \o (X \* Y \- fine 'E_P[X] \o* Y)%R).
by rewrite compreBr ?integrableB// compre_scale ?integrableZl.
rewrite expectationD/=; last 2 first.
- by rewrite compreBr// integrableB// compre_scale ?integrableZl.
- by rewrite compre_scale// integrableZl// finite_measure_integrable_cst.
rewrite 2?expectationB//= ?compre_scale// ?integrableZl//.
rewrite 3?expectationM//= ?finite_measure_integrable_cst//.
by rewrite expectation_cst mule1 fineM// EFinM !fineK// muleC subeK ?fin_numM.
Qed.
have ? : 'E_P[X] \is a fin_num by rewrite fin_num_abs// integrable_expectation.
have ? : 'E_P[Y] \is a fin_num by rewrite fin_num_abs// integrable_expectation.
rewrite unlock [X in 'E_P[X]](_ : _ = (X \* Y \- fine 'E_P[X] \o* Y
\- fine 'E_P[Y] \o* X \+ fine ('E_P[X] * 'E_P[Y]) \o* cst 1)%R); last first.
apply/funeqP => x /=; rewrite mulrDr !mulrDl/= mul1r fineM// mulrNN addrA.
by rewrite mulrN mulNr [Z in (X x * Y x - Z)%R]mulrC.
have ? : P.-integrable [set: T] (EFin \o (X \* Y \- fine 'E_P[X] \o* Y)%R).
by rewrite compreBr ?integrableB// compre_scale ?integrableZl.
rewrite expectationD/=; last 2 first.
- by rewrite compreBr// integrableB// compre_scale ?integrableZl.
- by rewrite compre_scale// integrableZl// finite_measure_integrable_cst.
rewrite 2?expectationB//= ?compre_scale// ?integrableZl//.
rewrite 3?expectationM//= ?finite_measure_integrable_cst//.
by rewrite expectation_cst mule1 fineM// EFinM !fineK// muleC subeK ?fin_numM.
Qed.
Lemma covarianceC (X Y : T -> R) : covariance P X Y = covariance P Y X.
Proof.
Lemma covariance_fin_num (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) ->
P.-integrable setT (EFin \o Y) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
covariance P X Y \is a fin_num.
Proof.
Lemma covariance_cst_l c (X : {RV P >-> R}) : covariance P (cst c) X = 0.
Proof.
Lemma covariance_cst_r (X : {RV P >-> R}) c : covariance P X (cst c) = 0.
Proof.
Lemma covarianceZl a (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) ->
P.-integrable setT (EFin \o Y) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
covariance P (a \o* X)%R Y = a%:E * covariance P X Y.
Proof.
move=> X1 Y1 XY1.
have aXY : (a \o* X * Y = a \o* (X * Y))%R.
by apply/funeqP => x; rewrite mulrAC.
rewrite [LHS]covarianceE => [||//|] /=; last 2 first.
- by rewrite compre_scale ?integrableZl.
- by rewrite aXY compre_scale ?integrableZl.
rewrite covarianceE// aXY !expectationM//.
by rewrite -muleA -muleBr// fin_num_adde_defr// expectation_fin_num.
Qed.
have aXY : (a \o* X * Y = a \o* (X * Y))%R.
by apply/funeqP => x; rewrite mulrAC.
rewrite [LHS]covarianceE => [||//|] /=; last 2 first.
- by rewrite compre_scale ?integrableZl.
- by rewrite aXY compre_scale ?integrableZl.
rewrite covarianceE// aXY !expectationM//.
by rewrite -muleA -muleBr// fin_num_adde_defr// expectation_fin_num.
Qed.
Lemma covarianceZr a (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) ->
P.-integrable setT (EFin \o Y) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
covariance P X (a \o* Y)%R = a%:E * covariance P X Y.
Proof.
Lemma covarianceNl (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) ->
P.-integrable setT (EFin \o Y) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
covariance P (\- X)%R Y = - covariance P X Y.
Proof.
Lemma covarianceNr (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) ->
P.-integrable setT (EFin \o Y) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
covariance P X (\- Y)%R = - covariance P X Y.
Proof.
Lemma covarianceNN (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) ->
P.-integrable setT (EFin \o Y) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
covariance P (\- X)%R (\- Y)%R = covariance P X Y.
Proof.
move=> X1 Y1 XY1.
have NY : P.-integrable setT (EFin \o (\- Y)%R) by rewrite compreN ?integrableN.
by rewrite covarianceNl ?covarianceNr ?oppeK//= mulrN compreN ?integrableN.
Qed.
have NY : P.-integrable setT (EFin \o (\- Y)%R) by rewrite compreN ?integrableN.
by rewrite covarianceNl ?covarianceNr ?oppeK//= mulrN compreN ?integrableN.
Qed.
Lemma covarianceDl (X Y Z : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
P.-integrable setT (EFin \o Z) -> P.-integrable setT (EFin \o (Z ^+ 2)%R) ->
P.-integrable setT (EFin \o (X * Z)%R) ->
P.-integrable setT (EFin \o (Y * Z)%R) ->
covariance P (X \+ Y)%R Z = covariance P X Z + covariance P Y Z.
Proof.
move=> X1 X2 Y1 Y2 Z1 Z2 XZ1 YZ1.
rewrite [LHS]covarianceE//= ?mulrDl ?compreDr// ?integrableD//.
rewrite 2?expectationD//=.
rewrite muleDl ?fin_num_adde_defr ?expectation_fin_num//.
rewrite oppeD ?fin_num_adde_defr ?fin_numM ?expectation_fin_num//.
by rewrite addeACA 2?covarianceE.
Qed.
rewrite [LHS]covarianceE//= ?mulrDl ?compreDr// ?integrableD//.
rewrite 2?expectationD//=.
rewrite muleDl ?fin_num_adde_defr ?expectation_fin_num//.
rewrite oppeD ?fin_num_adde_defr ?fin_numM ?expectation_fin_num//.
by rewrite addeACA 2?covarianceE.
Qed.
Lemma covarianceDr (X Y Z : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
P.-integrable setT (EFin \o Z) -> P.-integrable setT (EFin \o (Z ^+ 2)%R) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
P.-integrable setT (EFin \o (X * Z)%R) ->
covariance P X (Y \+ Z)%R = covariance P X Y + covariance P X Z.
Proof.
move=> X1 X2 Y1 Y2 Z1 Z2 XY1 XZ1.
by rewrite covarianceC covarianceDl ?(covarianceC X) 1?mulrC.
Qed.
by rewrite covarianceC covarianceDl ?(covarianceC X) 1?mulrC.
Qed.
Lemma covarianceBl (X Y Z : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
P.-integrable setT (EFin \o Z) -> P.-integrable setT (EFin \o (Z ^+ 2)%R) ->
P.-integrable setT (EFin \o (X * Z)%R) ->
P.-integrable setT (EFin \o (Y * Z)%R) ->
covariance P (X \- Y)%R Z = covariance P X Z - covariance P Y Z.
Proof.
move=> X1 X2 Y1 Y2 Z1 Z2 XZ1 YZ1.
rewrite -[(X \- Y)%R]/(X \+ (\- Y))%R covarianceDl ?covarianceNl//=.
- by rewrite compreN// integrableN.
- by rewrite mulrNN.
- by rewrite mulNr compreN// integrableN.
Qed.
rewrite -[(X \- Y)%R]/(X \+ (\- Y))%R covarianceDl ?covarianceNl//=.
- by rewrite compreN// integrableN.
- by rewrite mulrNN.
- by rewrite mulNr compreN// integrableN.
Qed.
Lemma covarianceBr (X Y Z : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
P.-integrable setT (EFin \o Z) -> P.-integrable setT (EFin \o (Z ^+ 2)%R) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
P.-integrable setT (EFin \o (X * Z)%R) ->
covariance P X (Y \- Z)%R = covariance P X Y - covariance P X Z.
Proof.
End covariance_lemmas.
Section variance.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Definition variance (X : T -> R) := covariance P X X.
Local Notation "''V_' P [ X ]" := (variance X).
Lemma varianceE (X : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
'V_P[X] = 'E_P[X ^+ 2] - ('E_P[X]) ^+ 2.
Proof.
Lemma variance_fin_num (X : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o X ^+ 2)%R ->
'V_P[X] \is a fin_num.
Proof.
Lemma variance_ge0 (X : {RV P >-> R}) : (0 <= 'V_P[X])%E.
Proof.
Lemma variance_cst r : 'V_P[cst r] = 0%E.
Proof.
Lemma varianceZ a (X : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
'V_P[(a \o* X)%R] = (a ^+ 2)%:E * 'V_P[X].
Proof.
move=> X1 X2; rewrite /variance covarianceZl//=.
- by rewrite covarianceZr// muleA.
- by rewrite compre_scale// integrableZl.
- rewrite [X in EFin \o X](_ : _ = (a \o* X ^+ 2)%R); last first.
by apply/funeqP => x; rewrite mulrA.
by rewrite compre_scale// integrableZl.
Qed.
- by rewrite covarianceZr// muleA.
- by rewrite compre_scale// integrableZl.
- rewrite [X in EFin \o X](_ : _ = (a \o* X ^+ 2)%R); last first.
by apply/funeqP => x; rewrite mulrA.
by rewrite compre_scale// integrableZl.
Qed.
Lemma varianceN (X : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
'V_P[(\- X)%R] = 'V_P[X].
Proof.
Lemma varianceD (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
'V_P[X \+ Y]%R = 'V_P[X] + 'V_P[Y] + 2%:E * covariance P X Y.
Proof.
move=> X1 X2 Y1 Y2 XY1.
rewrite -['V_P[_]]/(covariance P (X \+ Y)%R (X \+ Y)%R).
have XY : P.-integrable [set: T] (EFin \o (X \+ Y)%R).
by rewrite compreDr// integrableD.
rewrite covarianceDl//=; last 3 first.
- rewrite -expr2 sqrrD compreDr ?integrableD// compreDr// integrableD//.
rewrite -mulr_natr -[(_ * 2)%R]/(2 \o* (X * Y))%R compre_scale//.
exact: integrableZl.
- by rewrite mulrDr compreDr ?integrableD.
- by rewrite mulrDr mulrC compreDr ?integrableD.
rewrite covarianceDr// covarianceDr; [|by []..|by rewrite mulrC |exact: Y2].
rewrite (covarianceC P Y X) [LHS]addeA [LHS](ACl (1*4*(2*3)))/=.
by rewrite -[2%R]/(1 + 1)%R EFinD muleDl ?mul1e// covariance_fin_num.
Qed.
rewrite -['V_P[_]]/(covariance P (X \+ Y)%R (X \+ Y)%R).
have XY : P.-integrable [set: T] (EFin \o (X \+ Y)%R).
by rewrite compreDr// integrableD.
rewrite covarianceDl//=; last 3 first.
- rewrite -expr2 sqrrD compreDr ?integrableD// compreDr// integrableD//.
rewrite -mulr_natr -[(_ * 2)%R]/(2 \o* (X * Y))%R compre_scale//.
exact: integrableZl.
- by rewrite mulrDr compreDr ?integrableD.
- by rewrite mulrDr mulrC compreDr ?integrableD.
rewrite covarianceDr// covarianceDr; [|by []..|by rewrite mulrC |exact: Y2].
rewrite (covarianceC P Y X) [LHS]addeA [LHS](ACl (1*4*(2*3)))/=.
by rewrite -[2%R]/(1 + 1)%R EFinD muleDl ?mul1e// covariance_fin_num.
Qed.
Lemma varianceB (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
'V_P[(X \- Y)%R] = 'V_P[X] + 'V_P[Y] - 2%:E * covariance P X Y.
Proof.
move=> X1 X2 Y1 Y2 XY1.
rewrite -[(X \- Y)%R]/(X \+ (\- Y))%R.
rewrite varianceD/= ?varianceN ?covarianceNr ?muleN//.
- by rewrite compreN ?integrableN.
- by rewrite mulrNN.
- by rewrite mulrN compreN ?integrableN.
Qed.
rewrite -[(X \- Y)%R]/(X \+ (\- Y))%R.
rewrite varianceD/= ?varianceN ?covarianceNr ?muleN//.
- by rewrite compreN ?integrableN.
- by rewrite mulrNN.
- by rewrite mulrN compreN ?integrableN.
Qed.
Lemma varianceD_cst_l c (X : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
'V_P[(cst c \+ X)%R] = 'V_P[X].
Proof.
move=> X1 X2.
rewrite varianceD//=; last 3 first.
- exact: finite_measure_integrable_cst.
- by rewrite compre_scale// integrableZl// finite_measure_integrable_cst.
- by rewrite mulrC compre_scale ?integrableZl.
by rewrite variance_cst add0e covariance_cst_l mule0 adde0.
Qed.
rewrite varianceD//=; last 3 first.
- exact: finite_measure_integrable_cst.
- by rewrite compre_scale// integrableZl// finite_measure_integrable_cst.
- by rewrite mulrC compre_scale ?integrableZl.
by rewrite variance_cst add0e covariance_cst_l mule0 adde0.
Qed.
Lemma varianceD_cst_r (X : {RV P >-> R}) c :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
'V_P[(X \+ cst c)%R] = 'V_P[X].
Proof.
Lemma varianceB_cst_l c (X : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
'V_P[(cst c \- X)%R] = 'V_P[X].
Proof.
move=> X1 X2.
rewrite -[(cst c \- X)%R]/(cst c \+ (\- X))%R varianceD_cst_l/=; last 2 first.
- by rewrite compreN ?integrableN.
- by rewrite mulrNN; apply: X2.
by rewrite varianceN.
Qed.
rewrite -[(cst c \- X)%R]/(cst c \+ (\- X))%R varianceD_cst_l/=; last 2 first.
- by rewrite compreN ?integrableN.
- by rewrite mulrNN; apply: X2.
by rewrite varianceN.
Qed.
Lemma varianceB_cst_r (X : {RV P >-> R}) c :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
'V_P[(X \- cst c)%R] = 'V_P[X].
Lemma covariance_le (X Y : {RV P >-> R}) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
P.-integrable setT (EFin \o (X * Y)%R) ->
covariance P X Y <= sqrte 'V_P[X] * sqrte 'V_P[Y].
Proof.
move=> X1 X2 Y1 Y2 XY1.
rewrite -sqrteM ?variance_ge0//.
rewrite lee_sqrE ?sqrte_ge0// sqr_sqrte ?mule_ge0 ?variance_ge0//.
rewrite -(fineK (variance_fin_num X1 X2)) -(fineK (variance_fin_num Y1 Y2)).
rewrite -(fineK (covariance_fin_num X1 Y1 XY1)).
rewrite -EFin_expe -EFinM lee_fin -(@ler_pM2l _ 4) ?ltr0n// [leRHS]mulrA.
rewrite [in leLHS](_ : 4 = 2 * 2)%R -natrM// [in leLHS]natrM mulrACA -expr2.
rewrite -subr_le0; apply: deg_le2_ge0 => r; rewrite -lee_fin !EFinD.
rewrite EFinM fineK ?variance_fin_num// muleC -varianceZ//.
rewrite 2!EFinM ?fineK ?variance_fin_num// ?covariance_fin_num//.
rewrite -muleA [_ * r%:E]muleC -covarianceZl//.
rewrite addeAC -varianceD ?variance_ge0//=.
- by rewrite compre_scale ?integrableZl.
- rewrite [X in EFin \o X](_ : _ = r ^+2 \o* X ^+ 2)%R 1?mulrACA//.
by rewrite compre_scale ?integrableZl.
- by rewrite -mulrAC compre_scale// integrableZl.
Qed.
rewrite -sqrteM ?variance_ge0//.
rewrite lee_sqrE ?sqrte_ge0// sqr_sqrte ?mule_ge0 ?variance_ge0//.
rewrite -(fineK (variance_fin_num X1 X2)) -(fineK (variance_fin_num Y1 Y2)).
rewrite -(fineK (covariance_fin_num X1 Y1 XY1)).
rewrite -EFin_expe -EFinM lee_fin -(@ler_pM2l _ 4) ?ltr0n// [leRHS]mulrA.
rewrite [in leLHS](_ : 4 = 2 * 2)%R -natrM// [in leLHS]natrM mulrACA -expr2.
rewrite -subr_le0; apply: deg_le2_ge0 => r; rewrite -lee_fin !EFinD.
rewrite EFinM fineK ?variance_fin_num// muleC -varianceZ//.
rewrite 2!EFinM ?fineK ?variance_fin_num// ?covariance_fin_num//.
rewrite -muleA [_ * r%:E]muleC -covarianceZl//.
rewrite addeAC -varianceD ?variance_ge0//=.
- by rewrite compre_scale ?integrableZl.
- rewrite [X in EFin \o X](_ : _ = r ^+2 \o* X ^+ 2)%R 1?mulrACA//.
by rewrite compre_scale ?integrableZl.
- by rewrite -mulrAC compre_scale// integrableZl.
Qed.
End variance.
Notation "'V_ P [ X ]" := (variance P X).
Section markov_chebyshev_cantelli.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Lemma markov (X : {RV P >-> R}) (f : R -> R) (eps : R) :
(0 < eps)%R ->
measurable_fun [set: R] f -> (forall r, 0 <= r -> 0 <= f r)%R ->
{in Num.nneg &, {homo f : x y / x <= y}}%R ->
(f eps)%:E * P [set x | eps%:E <= `| (X x)%:E | ] <=
'E_P[f \o (fun x => `| x |%R) \o X].
Proof.
move=> e0 mf f0 f_nd; rewrite -(setTI [set _ | _]).
apply: (le_trans (@le_integral_comp_abse _ _ _ P _ measurableT (EFin \o X)
eps (er_map f) _ _ _ _ e0)) => //=.
- exact: measurable_er_map.
- by case => //= r _; exact: f0.
- move=> [x| |] [y| |]; rewrite !inE/= !in_itv/= ?andbT ?lee_fin ?leey//.
by move=> ? ? ?; rewrite f_nd.
- exact/EFin_measurable_fun.
- by rewrite unlock.
Qed.
apply: (le_trans (@le_integral_comp_abse _ _ _ P _ measurableT (EFin \o X)
eps (er_map f) _ _ _ _ e0)) => //=.
- exact: measurable_er_map.
- by case => //= r _; exact: f0.
- move=> [x| |] [y| |]; rewrite !inE/= !in_itv/= ?andbT ?lee_fin ?leey//.
by move=> ? ? ?; rewrite f_nd.
- exact/EFin_measurable_fun.
- by rewrite unlock.
Qed.
Definition mmt_gen_fun (X : {RV P >-> R}) (t : R) := 'E_P[expR \o t \o* X].
Lemma chernoff (X : {RV P >-> R}) (r a : R) : (0 < r)%R ->
P [set x | X x >= a]%R <= mmt_gen_fun X r * (expR (- (r * a)))%:E.
Proof.
move=> t0.
rewrite /mmt_gen_fun; have -> : expR \o r \o* X =
(normr \o normr) \o [the {mfun T >-> R} of expR \o r \o* X].
by apply: funext => t /=; rewrite normr_id ger0_norm ?expR_ge0.
rewrite expRN lee_pdivlMr ?expR_gt0//.
rewrite (le_trans _ (markov _ (expR_gt0 (r * a)) _ _ _))//; last first.
exact: (monoW_in (@ger0_le_norm _)).
rewrite ger0_norm ?expR_ge0// muleC lee_pmul2l// ?lte_fin ?expR_gt0//.
rewrite [X in _ <= P X](_ : _ = [set x | a <= X x]%R)//; apply: eq_set => t/=.
by rewrite ger0_norm ?expR_ge0// lee_fin ler_expR mulrC ler_pM2r.
Qed.
rewrite /mmt_gen_fun; have -> : expR \o r \o* X =
(normr \o normr) \o [the {mfun T >-> R} of expR \o r \o* X].
by apply: funext => t /=; rewrite normr_id ger0_norm ?expR_ge0.
rewrite expRN lee_pdivlMr ?expR_gt0//.
rewrite (le_trans _ (markov _ (expR_gt0 (r * a)) _ _ _))//; last first.
exact: (monoW_in (@ger0_le_norm _)).
rewrite ger0_norm ?expR_ge0// muleC lee_pmul2l// ?lte_fin ?expR_gt0//.
rewrite [X in _ <= P X](_ : _ = [set x | a <= X x]%R)//; apply: eq_set => t/=.
by rewrite ger0_norm ?expR_ge0// lee_fin ler_expR mulrC ler_pM2r.
Qed.
Lemma chebyshev (X : {RV P >-> R}) (eps : R) : (0 < eps)%R ->
P [set x | (eps <= `| X x - fine ('E_P[X])|)%R ] <= (eps ^- 2)%:E * 'V_P[X].
Proof.
move => heps; have [->|hv] := eqVneq 'V_P[X] +oo.
by rewrite mulr_infty gtr0_sg ?mul1e// ?leey// invr_gt0// exprn_gt0.
have h (Y : {RV P >-> R}) :
P [set x | (eps <= `|Y x|)%R] <= (eps ^- 2)%:E * 'E_P[Y ^+ 2].
rewrite -lee_pdivrMl; last by rewrite invr_gt0// exprn_gt0.
rewrite exprnN expfV exprz_inv opprK -exprnP.
apply: (@le_trans _ _ ('E_P[(@GRing.exp R ^~ 2%N \o normr) \o Y])).
apply: (@markov Y (@GRing.exp R ^~ 2%N)) => //.
- by move=> r _; exact: sqr_ge0.
- move=> x y; rewrite !nnegrE => x0 y0.
by rewrite ler_sqr.
apply: expectation_le => //.
- by apply: measurableT_comp => //; exact: measurableT_comp.
- by move=> x /=; exact: sqr_ge0.
- by move=> x /=; exact: sqr_ge0.
- by apply/aeW => t /=; rewrite real_normK// num_real.
have := h [the {mfun T >-> R} of (X \- cst (fine ('E_P[X])))%R].
by move=> /le_trans; apply; rewrite /variance [in leRHS]unlock.
Qed.
by rewrite mulr_infty gtr0_sg ?mul1e// ?leey// invr_gt0// exprn_gt0.
have h (Y : {RV P >-> R}) :
P [set x | (eps <= `|Y x|)%R] <= (eps ^- 2)%:E * 'E_P[Y ^+ 2].
rewrite -lee_pdivrMl; last by rewrite invr_gt0// exprn_gt0.
rewrite exprnN expfV exprz_inv opprK -exprnP.
apply: (@le_trans _ _ ('E_P[(@GRing.exp R ^~ 2%N \o normr) \o Y])).
apply: (@markov Y (@GRing.exp R ^~ 2%N)) => //.
- by move=> r _; exact: sqr_ge0.
- move=> x y; rewrite !nnegrE => x0 y0.
by rewrite ler_sqr.
apply: expectation_le => //.
- by apply: measurableT_comp => //; exact: measurableT_comp.
- by move=> x /=; exact: sqr_ge0.
- by move=> x /=; exact: sqr_ge0.
- by apply/aeW => t /=; rewrite real_normK// num_real.
have := h [the {mfun T >-> R} of (X \- cst (fine ('E_P[X])))%R].
by move=> /le_trans; apply; rewrite /variance [in leRHS]unlock.
Qed.
Lemma cantelli (X : {RV P >-> R}) (lambda : R) :
P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
(0 < lambda)%R ->
P [set x | lambda%:E <= (X x)%:E - 'E_P[X]]
<= (fine 'V_P[X] / (fine 'V_P[X] + lambda^2))%:E.
Proof.
move=> X1 X2 lambda_gt0.
have finEK : (fine 'E_P[X])%:E = 'E_P[X].
by rewrite fineK ?unlock ?integral_fune_fin_num.
have finVK : (fine 'V_P[X])%:E = 'V_P[X] by rewrite fineK ?variance_fin_num.
pose Y := (X \- cst (fine 'E_P[X]))%R.
have Y1 : P.-integrable [set: T] (EFin \o Y).
rewrite compreBr => [|//]; apply: integrableB X1 _ => [//|].
exact: finite_measure_integrable_cst.
have Y2 : P.-integrable [set: T] (EFin \o (Y ^+ 2)%R).
rewrite sqrrD/= compreDr => [|//].
apply: integrableD => [//||]; last first.
rewrite -[(_ ^+ 2)%R]/(cst ((- fine 'E_P[X]) ^+ 2)%R).
exact: finite_measure_integrable_cst.
rewrite compreDr => [|//]; apply: integrableD X2 _ => [//|].
rewrite [X in EFin \o X](_ : _ = (- fine 'E_P[X] * 2) \o* X)%R; last first.
by apply/funeqP => x /=; rewrite -mulr_natl mulrC mulrA.
by rewrite compre_scale => [|//]; apply: integrableZl X1.
have EY : 'E_P[Y] = 0.
rewrite expectationB/= ?finite_measure_integrable_cst//.
rewrite expectation_cst finEK subee//.
by rewrite unlock; apply: integral_fune_fin_num X1.
have VY : 'V_P[Y] = 'V_P[X] by rewrite varianceB_cst_r.
have le (u : R) : (0 <= u)%R ->
P [set x | lambda%:E <= (X x)%:E - 'E_P[X]]
<= ((fine 'V_P[X] + u^2) / (lambda + u)^2)%:E.
move=> uge0; rewrite EFinM.
have YU1 : P.-integrable [set: T] (EFin \o (Y \+ cst u)%R).
rewrite compreDr => [|//]; apply: integrableD Y1 _ => [//|].
exact: finite_measure_integrable_cst.
have YU2 : P.-integrable [set: T] (EFin \o ((Y \+ cst u) ^+ 2)%R).
rewrite sqrrD/= compreDr => [|//].
apply: integrableD => [//||]; last first.
rewrite -[(_ ^+ 2)%R]/(cst (u ^+ 2))%R.
exact: finite_measure_integrable_cst.
rewrite compreDr => [|//]; apply: integrableD Y2 _ => [//|].
rewrite [X in EFin \o X](_ : _ = (2 * u) \o* Y)%R; last first.
by apply/funeqP => x /=; rewrite -mulr_natl mulrCA.
by rewrite compre_scale => [|//]; apply: integrableZl Y1.
have -> : (fine 'V_P[X] + u^2)%:E = 'E_P[(Y \+ cst u)^+2]%R.
rewrite -VY -[RHS](@subeK _ _ (('E_P[(Y \+ cst u)%R])^+2)); last first.
by rewrite fin_numX ?unlock ?integral_fune_fin_num.
rewrite -varianceE/= -/Y -?expe2//.
rewrite expectationD/= ?EY ?add0e ?expectation_cst -?EFinM; last 2 first.
- rewrite compreBr => [|//]; apply: integrableB X1 _ => [//|].
exact: finite_measure_integrable_cst.
- exact: finite_measure_integrable_cst.
by rewrite (varianceD_cst_r _ Y1 Y2) EFinD fineK ?(variance_fin_num Y1 Y2).
have le : [set x | lambda%:E <= (X x)%:E - 'E_P[X]]
`<=` [set x | ((lambda + u)^2)%:E <= ((Y x + u)^+2)%:E].
move=> x /= le; rewrite lee_fin; apply: lerXn2r.
- exact: addr_ge0 (ltW lambda_gt0) _.
- apply/(addr_ge0 _ uge0)/(le_trans (ltW lambda_gt0) _).
by rewrite -lee_fin EFinB finEK.
- by rewrite lerD2r -lee_fin EFinB finEK.
apply: (le_trans (le_measure _ _ _ le)).
- rewrite -[[set _ | _]]setTI inE; apply: emeasurable_fun_c_infty => [//|].
by apply: emeasurable_funB => //; exact: measurable_int X1.
- rewrite -[[set _ | _]]setTI inE; apply: emeasurable_fun_c_infty => [//|].
rewrite EFin_measurable_fun [X in measurable_fun _ X](_ : _ =
(fun x => x ^+ 2) \o (fun x => Y x + u))%R//.
by apply/measurableT_comp => //; apply/measurable_funD.
set eps := ((lambda + u) ^ 2)%R.
have peps : (0 < eps)%R by rewrite exprz_gt0 ?ltr_wpDr.
rewrite (lee_pdivlMr _ _ peps) muleC.
under eq_set => x.
rewrite -[leRHS]gee0_abs ?lee_fin ?sqr_ge0 -?lee_fin => [|//].
rewrite -[(_ ^+ 2)%R]/(((Y \+ cst u) ^+ 2) x)%R; over.
rewrite -[X in X%:E * _]gtr0_norm => [|//].
apply: (le_trans (markov _ peps _ _ _)) => //=.
by move=> x y /[!nnegrE] /ger0_norm-> /ger0_norm->.
rewrite -/Y le_eqVlt; apply/orP; left; apply/eqP; congr expectation.
by apply/funeqP => x /=; rewrite -expr2 normr_id ger0_norm ?sqr_ge0.
pose u0 := (fine 'V_P[X] / lambda)%R.
have u0ge0 : (0 <= u0)%R.
by apply: divr_ge0 (ltW lambda_gt0); rewrite -lee_fin finVK variance_ge0.
apply: le_trans (le _ u0ge0) _; rewrite lee_fin le_eqVlt; apply/orP; left.
rewrite eqr_div; [|apply: lt0r_neq0..]; last 2 first.
- by rewrite exprz_gt0 -1?[ltLHS]addr0 ?ltr_leD.
- by rewrite ltr_wpDl ?fine_ge0 ?variance_ge0 ?exprz_gt0.
apply/eqP; have -> : fine 'V_P[X] = (u0 * lambda)%R.
by rewrite /u0 -mulrA mulVr ?mulr1 ?unitfE ?gt_eqF.
by rewrite -mulrDl -mulrDr (addrC u0) [in RHS](mulrAC u0) -exprnP expr2 !mulrA.
Qed.
have finEK : (fine 'E_P[X])%:E = 'E_P[X].
by rewrite fineK ?unlock ?integral_fune_fin_num.
have finVK : (fine 'V_P[X])%:E = 'V_P[X] by rewrite fineK ?variance_fin_num.
pose Y := (X \- cst (fine 'E_P[X]))%R.
have Y1 : P.-integrable [set: T] (EFin \o Y).
rewrite compreBr => [|//]; apply: integrableB X1 _ => [//|].
exact: finite_measure_integrable_cst.
have Y2 : P.-integrable [set: T] (EFin \o (Y ^+ 2)%R).
rewrite sqrrD/= compreDr => [|//].
apply: integrableD => [//||]; last first.
rewrite -[(_ ^+ 2)%R]/(cst ((- fine 'E_P[X]) ^+ 2)%R).
exact: finite_measure_integrable_cst.
rewrite compreDr => [|//]; apply: integrableD X2 _ => [//|].
rewrite [X in EFin \o X](_ : _ = (- fine 'E_P[X] * 2) \o* X)%R; last first.
by apply/funeqP => x /=; rewrite -mulr_natl mulrC mulrA.
by rewrite compre_scale => [|//]; apply: integrableZl X1.
have EY : 'E_P[Y] = 0.
rewrite expectationB/= ?finite_measure_integrable_cst//.
rewrite expectation_cst finEK subee//.
by rewrite unlock; apply: integral_fune_fin_num X1.
have VY : 'V_P[Y] = 'V_P[X] by rewrite varianceB_cst_r.
have le (u : R) : (0 <= u)%R ->
P [set x | lambda%:E <= (X x)%:E - 'E_P[X]]
<= ((fine 'V_P[X] + u^2) / (lambda + u)^2)%:E.
move=> uge0; rewrite EFinM.
have YU1 : P.-integrable [set: T] (EFin \o (Y \+ cst u)%R).
rewrite compreDr => [|//]; apply: integrableD Y1 _ => [//|].
exact: finite_measure_integrable_cst.
have YU2 : P.-integrable [set: T] (EFin \o ((Y \+ cst u) ^+ 2)%R).
rewrite sqrrD/= compreDr => [|//].
apply: integrableD => [//||]; last first.
rewrite -[(_ ^+ 2)%R]/(cst (u ^+ 2))%R.
exact: finite_measure_integrable_cst.
rewrite compreDr => [|//]; apply: integrableD Y2 _ => [//|].
rewrite [X in EFin \o X](_ : _ = (2 * u) \o* Y)%R; last first.
by apply/funeqP => x /=; rewrite -mulr_natl mulrCA.
by rewrite compre_scale => [|//]; apply: integrableZl Y1.
have -> : (fine 'V_P[X] + u^2)%:E = 'E_P[(Y \+ cst u)^+2]%R.
rewrite -VY -[RHS](@subeK _ _ (('E_P[(Y \+ cst u)%R])^+2)); last first.
by rewrite fin_numX ?unlock ?integral_fune_fin_num.
rewrite -varianceE/= -/Y -?expe2//.
rewrite expectationD/= ?EY ?add0e ?expectation_cst -?EFinM; last 2 first.
- rewrite compreBr => [|//]; apply: integrableB X1 _ => [//|].
exact: finite_measure_integrable_cst.
- exact: finite_measure_integrable_cst.
by rewrite (varianceD_cst_r _ Y1 Y2) EFinD fineK ?(variance_fin_num Y1 Y2).
have le : [set x | lambda%:E <= (X x)%:E - 'E_P[X]]
`<=` [set x | ((lambda + u)^2)%:E <= ((Y x + u)^+2)%:E].
move=> x /= le; rewrite lee_fin; apply: lerXn2r.
- exact: addr_ge0 (ltW lambda_gt0) _.
- apply/(addr_ge0 _ uge0)/(le_trans (ltW lambda_gt0) _).
by rewrite -lee_fin EFinB finEK.
- by rewrite lerD2r -lee_fin EFinB finEK.
apply: (le_trans (le_measure _ _ _ le)).
- rewrite -[[set _ | _]]setTI inE; apply: emeasurable_fun_c_infty => [//|].
by apply: emeasurable_funB => //; exact: measurable_int X1.
- rewrite -[[set _ | _]]setTI inE; apply: emeasurable_fun_c_infty => [//|].
rewrite EFin_measurable_fun [X in measurable_fun _ X](_ : _ =
(fun x => x ^+ 2) \o (fun x => Y x + u))%R//.
by apply/measurableT_comp => //; apply/measurable_funD.
set eps := ((lambda + u) ^ 2)%R.
have peps : (0 < eps)%R by rewrite exprz_gt0 ?ltr_wpDr.
rewrite (lee_pdivlMr _ _ peps) muleC.
under eq_set => x.
rewrite -[leRHS]gee0_abs ?lee_fin ?sqr_ge0 -?lee_fin => [|//].
rewrite -[(_ ^+ 2)%R]/(((Y \+ cst u) ^+ 2) x)%R; over.
rewrite -[X in X%:E * _]gtr0_norm => [|//].
apply: (le_trans (markov _ peps _ _ _)) => //=.
by move=> x y /[!nnegrE] /ger0_norm-> /ger0_norm->.
rewrite -/Y le_eqVlt; apply/orP; left; apply/eqP; congr expectation.
by apply/funeqP => x /=; rewrite -expr2 normr_id ger0_norm ?sqr_ge0.
pose u0 := (fine 'V_P[X] / lambda)%R.
have u0ge0 : (0 <= u0)%R.
by apply: divr_ge0 (ltW lambda_gt0); rewrite -lee_fin finVK variance_ge0.
apply: le_trans (le _ u0ge0) _; rewrite lee_fin le_eqVlt; apply/orP; left.
rewrite eqr_div; [|apply: lt0r_neq0..]; last 2 first.
- by rewrite exprz_gt0 -1?[ltLHS]addr0 ?ltr_leD.
- by rewrite ltr_wpDl ?fine_ge0 ?variance_ge0 ?exprz_gt0.
apply/eqP; have -> : fine 'V_P[X] = (u0 * lambda)%R.
by rewrite /u0 -mulrA mulVr ?mulr1 ?unitfE ?gt_eqF.
by rewrite -mulrDl -mulrDr (addrC u0) [in RHS](mulrAC u0) -exprnP expr2 !mulrA.
Qed.
End markov_chebyshev_cantelli.
HB.mixin Record MeasurableFun_isDiscrete d (T : measurableType d) (R : realType)
(X : T -> R) of @MeasurableFun d T R X := {
countable_range : countable (range X)
}.
HB.structure Definition discreteMeasurableFun d (T : measurableType d)
(R : realType) := {
X of isMeasurableFun d T R X & MeasurableFun_isDiscrete d T R X
}.
Notation "{ 'dmfun' aT >-> T }" :=
(@discreteMeasurableFun.type _ aT T) : form_scope.
Definition discrete_random_variable (d : _) (T : measurableType d)
(R : realType) (P : probability T R) := {dmfun T >-> R}.
Notation "{ 'dRV' P >-> R }" :=
(@discrete_random_variable _ _ R P) : form_scope.
Section dRV_definitions.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Definition dRV_dom_enum (X : {dRV P >-> R}) :
{ B : set nat & {splitbij B >-> range X}}.
Proof.
have /countable_bijP/cid[B] := @countable_range _ _ _ X.
move/card_esym/ppcard_eqP/unsquash => f.
exists B; exact: f.
Qed.
move/card_esym/ppcard_eqP/unsquash => f.
exists B; exact: f.
Qed.
Definition dRV_dom (X : {dRV P >-> R}) : set nat := projT1 (dRV_dom_enum X).
Definition dRV_enum (X : {dRV P >-> R}) : {splitbij (dRV_dom X) >-> range X} :=
projT2 (dRV_dom_enum X).
Definition enum_prob (X : {dRV P >-> R}) :=
(fun k => P (X @^-1` [set dRV_enum X k])) \_ (dRV_dom X).
End dRV_definitions.
Section distribution_dRV.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Variable X : {dRV P >-> R}.
Lemma distribution_dRV_enum (n : nat) : n \in dRV_dom X ->
distribution P X [set dRV_enum X n] = enum_prob X n.
Proof.
Lemma distribution_dRV A : measurable A ->
distribution P X A = \sum_(k <oo) enum_prob X k * \d_ (dRV_enum X k) A.
Proof.
move=> mA; rewrite /distribution /pushforward.
have mAX i : dRV_dom X i -> measurable (X @^-1` (A `&` [set dRV_enum X i])).
move=> _; rewrite preimage_setI; apply: measurableI => //.
exact/measurable_sfunP.
have tAX : trivIset (dRV_dom X) (fun k => X @^-1` (A `&` [set dRV_enum X k])).
under eq_fun do rewrite preimage_setI; rewrite -/(trivIset _ _).
apply: trivIset_setIl; apply/trivIsetP => i j iX jX /eqP ij.
rewrite -preimage_setI (_ : _ `&` _ = set0)//.
by apply/seteqP; split => //= x [] -> {x} /inj; rewrite inE inE => /(_ iX jX).
have := measure_bigcup P _ (fun k => X @^-1` (A `&` [set dRV_enum X k])) mAX tAX.
rewrite -preimage_bigcup => {mAX tAX}PXU.
rewrite -{1}(setIT A) -(setUv (\bigcup_(i in dRV_dom X) [set dRV_enum X i])).
rewrite setIUr preimage_setU measureU; last 3 first.
- rewrite preimage_setI; apply: measurableI => //.
exact: measurable_sfunP.
by apply: measurable_sfunP; exact: bigcup_measurable.
- apply: measurable_sfunP; apply: measurableI => //.
by apply: measurableC; exact: bigcup_measurable.
- rewrite 2!preimage_setI setIACA -!setIA -preimage_setI.
by rewrite setICr preimage_set0 2!setI0.
rewrite [X in _ + X = _](_ : _ = 0) ?adde0; last first.
rewrite (_ : _ @^-1` _ = set0) ?measure0//; apply/disjoints_subset => x AXx.
rewrite setCK /bigcup /=; exists ((dRV_enum X)^-1 (X x))%function.
exact: funS.
by rewrite invK// inE.
rewrite setI_bigcupr; etransitivity; first exact: PXU.
rewrite eseries_mkcond; apply: eq_eseriesr => k _.
rewrite /enum_prob patchE; case: ifPn => nX; rewrite ?mul0e//.
rewrite diracE; have [kA|] := boolP (_ \in A).
by rewrite mule1 setIidr// => _ /= ->; exact: set_mem.
rewrite notin_setE => kA.
rewrite mule0 (disjoints_subset _ _).2 ?preimage_set0 ?measure0//.
by apply: subsetCr; rewrite sub1set inE.
Qed.
have mAX i : dRV_dom X i -> measurable (X @^-1` (A `&` [set dRV_enum X i])).
move=> _; rewrite preimage_setI; apply: measurableI => //.
exact/measurable_sfunP.
have tAX : trivIset (dRV_dom X) (fun k => X @^-1` (A `&` [set dRV_enum X k])).
under eq_fun do rewrite preimage_setI; rewrite -/(trivIset _ _).
apply: trivIset_setIl; apply/trivIsetP => i j iX jX /eqP ij.
rewrite -preimage_setI (_ : _ `&` _ = set0)//.
by apply/seteqP; split => //= x [] -> {x} /inj; rewrite inE inE => /(_ iX jX).
have := measure_bigcup P _ (fun k => X @^-1` (A `&` [set dRV_enum X k])) mAX tAX.
rewrite -preimage_bigcup => {mAX tAX}PXU.
rewrite -{1}(setIT A) -(setUv (\bigcup_(i in dRV_dom X) [set dRV_enum X i])).
rewrite setIUr preimage_setU measureU; last 3 first.
- rewrite preimage_setI; apply: measurableI => //.
exact: measurable_sfunP.
by apply: measurable_sfunP; exact: bigcup_measurable.
- apply: measurable_sfunP; apply: measurableI => //.
by apply: measurableC; exact: bigcup_measurable.
- rewrite 2!preimage_setI setIACA -!setIA -preimage_setI.
by rewrite setICr preimage_set0 2!setI0.
rewrite [X in _ + X = _](_ : _ = 0) ?adde0; last first.
rewrite (_ : _ @^-1` _ = set0) ?measure0//; apply/disjoints_subset => x AXx.
rewrite setCK /bigcup /=; exists ((dRV_enum X)^-1 (X x))%function.
exact: funS.
by rewrite invK// inE.
rewrite setI_bigcupr; etransitivity; first exact: PXU.
rewrite eseries_mkcond; apply: eq_eseriesr => k _.
rewrite /enum_prob patchE; case: ifPn => nX; rewrite ?mul0e//.
rewrite diracE; have [kA|] := boolP (_ \in A).
by rewrite mule1 setIidr// => _ /= ->; exact: set_mem.
rewrite notin_setE => kA.
rewrite mule0 (disjoints_subset _ _).2 ?preimage_set0 ?measure0//.
by apply: subsetCr; rewrite sub1set inE.
Qed.
Lemma sum_enum_prob : \sum_(n <oo) (enum_prob X) n = 1.
Proof.
have := distribution_dRV measurableT.
rewrite probability_setT/= => /esym; apply: eq_trans.
by rewrite [RHS]eseries_mkcond; apply: eq_eseriesr => k _; rewrite diracT mule1.
Qed.
rewrite probability_setT/= => /esym; apply: eq_trans.
by rewrite [RHS]eseries_mkcond; apply: eq_eseriesr => k _; rewrite diracT mule1.
Qed.
End distribution_dRV.
Section discrete_distribution.
Local Open Scope ereal_scope.
Context d (T : measurableType d) (R : realType) (P : probability T R).
Lemma dRV_expectation (X : {dRV P >-> R}) :
P.-integrable [set: T] (EFin \o X) ->
'E_P[X] = \sum_(n <oo) enum_prob X n * (dRV_enum X n)%:E.
Proof.
move=> ix; rewrite unlock.
rewrite -[in LHS](_ : \bigcup_k (if k \in dRV_dom X then
X @^-1` [set dRV_enum X k] else set0) = setT); last first.
apply/seteqP; split => // t _.
exists ((dRV_enum X)^-1%function (X t)) => //.
case: ifPn=> [_|].
by rewrite invK// inE.
by rewrite notin_setE/=; apply; apply: funS.
have tA : trivIset (dRV_dom X) (fun k => [set dRV_enum X k]).
by move=> i j iX jX [r [/= ->{r}]] /inj; rewrite !inE; exact.
have {tA}/trivIset_mkcond tXA :
trivIset (dRV_dom X) (fun k => X @^-1` [set dRV_enum X k]).
apply/trivIsetP => /= i j iX jX ij.
move/trivIsetP : tA => /(_ i j iX jX) Aij.
by rewrite -preimage_setI Aij ?preimage_set0.
rewrite integral_bigcup //; last 2 first.
- by move=> k; case: ifPn.
- apply: (integrableS measurableT) => //.
by rewrite -bigcup_mkcond; exact: bigcup_measurable.
transitivity (\sum_(i <oo)
\int[P]_(x in (if i \in dRV_dom X then X @^-1` [set dRV_enum X i] else set0))
(dRV_enum X i)%:E).
apply: eq_eseriesr => i _; case: ifPn => iX.
by apply: eq_integral => t; rewrite in_setE/= => ->.
by rewrite !integral_set0.
transitivity (\sum_(i <oo) (dRV_enum X i)%:E *
\int[P]_(x in (if i \in dRV_dom X then X @^-1` [set dRV_enum X i] else set0))
1).
apply: eq_eseriesr => i _; rewrite -integralZl//; last 2 first.
- by case: ifPn.
- apply/integrableP; split => //.
rewrite (eq_integral (cst 1%E)); last by move=> x _; rewrite abse1.
rewrite integral_cst//; last by case: ifPn.
rewrite mul1e (@le_lt_trans _ _ 1%E) ?ltey//.
by case: ifPn => // _; exact: probability_le1.
by apply: eq_integral => y _; rewrite mule1.
apply: eq_eseriesr => k _; case: ifPn => kX.
rewrite /= integral_cst//= mul1e probability_distribution muleC.
by rewrite distribution_dRV_enum.
by rewrite integral_set0 mule0 /enum_prob patchE (negbTE kX) mul0e.
Qed.
rewrite -[in LHS](_ : \bigcup_k (if k \in dRV_dom X then
X @^-1` [set dRV_enum X k] else set0) = setT); last first.
apply/seteqP; split => // t _.
exists ((dRV_enum X)^-1%function (X t)) => //.
case: ifPn=> [_|].
by rewrite invK// inE.
by rewrite notin_setE/=; apply; apply: funS.
have tA : trivIset (dRV_dom X) (fun k => [set dRV_enum X k]).
by move=> i j iX jX [r [/= ->{r}]] /inj; rewrite !inE; exact.
have {tA}/trivIset_mkcond tXA :
trivIset (dRV_dom X) (fun k => X @^-1` [set dRV_enum X k]).
apply/trivIsetP => /= i j iX jX ij.
move/trivIsetP : tA => /(_ i j iX jX) Aij.
by rewrite -preimage_setI Aij ?preimage_set0.
rewrite integral_bigcup //; last 2 first.
- by move=> k; case: ifPn.
- apply: (integrableS measurableT) => //.
by rewrite -bigcup_mkcond; exact: bigcup_measurable.
transitivity (\sum_(i <oo)
\int[P]_(x in (if i \in dRV_dom X then X @^-1` [set dRV_enum X i] else set0))
(dRV_enum X i)%:E).
apply: eq_eseriesr => i _; case: ifPn => iX.
by apply: eq_integral => t; rewrite in_setE/= => ->.
by rewrite !integral_set0.
transitivity (\sum_(i <oo) (dRV_enum X i)%:E *
\int[P]_(x in (if i \in dRV_dom X then X @^-1` [set dRV_enum X i] else set0))
1).
apply: eq_eseriesr => i _; rewrite -integralZl//; last 2 first.
- by case: ifPn.
- apply/integrableP; split => //.
rewrite (eq_integral (cst 1%E)); last by move=> x _; rewrite abse1.
rewrite integral_cst//; last by case: ifPn.
rewrite mul1e (@le_lt_trans _ _ 1%E) ?ltey//.
by case: ifPn => // _; exact: probability_le1.
by apply: eq_integral => y _; rewrite mule1.
apply: eq_eseriesr => k _; case: ifPn => kX.
rewrite /= integral_cst//= mul1e probability_distribution muleC.
by rewrite distribution_dRV_enum.
by rewrite integral_set0 mule0 /enum_prob patchE (negbTE kX) mul0e.
Qed.
Definition pmf (X : {RV P >-> R}) (r : R) : R := fine (P (X @^-1` [set r])).
Lemma expectation_pmf (X : {dRV P >-> R}) :
P.-integrable [set: T] (EFin \o X) -> 'E_P[X] =
\sum_(n <oo | n \in dRV_dom X) (pmf X (dRV_enum X n))%:E * (dRV_enum X n)%:E.
Proof.
move=> iX; rewrite dRV_expectation// [in RHS]eseries_mkcond.
apply: eq_eseriesr => k _.
rewrite /enum_prob patchE; case: ifPn => kX; last by rewrite mul0e.
by rewrite /pmf fineK// fin_num_measure.
Qed.
apply: eq_eseriesr => k _.
rewrite /enum_prob patchE; case: ifPn => kX; last by rewrite mul0e.
by rewrite /pmf fineK// fin_num_measure.
Qed.
End discrete_distribution.
Section bernoulli_pmf.
Context {R : realType} (p : R).
Local Open Scope ring_scope.
Definition bernoulli_pmf b := if b then p else 1 - p.
Lemma bernoulli_pmf_ge0 (p01 : 0 <= p <= 1) b : 0 <= bernoulli_pmf b.
Proof.
Lemma bernoulli_pmf1 (p01 : 0 <= p <= 1) :
\sum_(i \in [set: bool]) (bernoulli_pmf i)%:E = 1%E.
Proof.
End bernoulli_pmf.
Lemma measurable_bernoulli_pmf {R : realType} D n :
measurable_fun D (@bernoulli_pmf R ^~ n).
Proof.
Definition bernoulli {R : realType} (p : R) : set bool -> \bar R := fun A =>
if (0 <= p <= 1)%R then \sum_(b \in A) (bernoulli_pmf p b)%:E else \d_false A.
Section bernoulli.
Context {R : realType} (p : R).
Local Notation bernoulli := (bernoulli p).
Let bernoulli0 : bernoulli set0 = 0.
Proof.
Let bernoulli_ge0 U : (0 <= bernoulli U)%E.
Proof.
rewrite /bernoulli; case: ifPn => // p01.
rewrite fsbig_finite//= sumEFin lee_fin.
by apply: sumr_ge0 => /= b _; exact: bernoulli_pmf_ge0.
Qed.
rewrite fsbig_finite//= sumEFin lee_fin.
by apply: sumr_ge0 => /= b _; exact: bernoulli_pmf_ge0.
Qed.
Let bernoulli_sigma_additive : semi_sigma_additive bernoulli.
Proof.
move=> F mF tF mUF; rewrite /bernoulli; case: ifPn => p01; last first.
exact: measure_semi_sigma_additive.
apply: cvg_toP.
apply: ereal_nondecreasing_is_cvgn => m n mn.
apply: lee_sum_nneg_natr => // k _ _.
rewrite fsbig_finite//= sumEFin lee_fin.
by apply: sumr_ge0 => /= b _; exact: bernoulli_pmf_ge0.
transitivity (\sum_(0 <= i <oo) (\esum_(j in F i) (bernoulli_pmf p j)%:E)%R)%E.
apply: eq_eseriesr => k _; rewrite esum_fset//= => b _.
by rewrite lee_fin bernoulli_pmf_ge0.
rewrite -nneseries_sum_bigcup//=; last first.
by move=> b; rewrite lee_fin bernoulli_pmf_ge0.
by rewrite esum_fset//= => b _; rewrite lee_fin bernoulli_pmf_ge0.
Qed.
exact: measure_semi_sigma_additive.
apply: cvg_toP.
apply: ereal_nondecreasing_is_cvgn => m n mn.
apply: lee_sum_nneg_natr => // k _ _.
rewrite fsbig_finite//= sumEFin lee_fin.
by apply: sumr_ge0 => /= b _; exact: bernoulli_pmf_ge0.
transitivity (\sum_(0 <= i <oo) (\esum_(j in F i) (bernoulli_pmf p j)%:E)%R)%E.
apply: eq_eseriesr => k _; rewrite esum_fset//= => b _.
by rewrite lee_fin bernoulli_pmf_ge0.
rewrite -nneseries_sum_bigcup//=; last first.
by move=> b; rewrite lee_fin bernoulli_pmf_ge0.
by rewrite esum_fset//= => b _; rewrite lee_fin bernoulli_pmf_ge0.
Qed.
HB.instance Definition _ := isMeasure.Build _ _ _ bernoulli
bernoulli0 bernoulli_ge0 bernoulli_sigma_additive.
Let bernoulli_setT : bernoulli [set: _] = 1%E.
Proof.
rewrite /bernoulli/=; case: ifPn => p01; last by rewrite probability_setT.
by rewrite bernoulli_pmf1.
Qed.
by rewrite bernoulli_pmf1.
Qed.
HB.instance Definition _ :=
@Measure_isProbability.Build _ _ R bernoulli bernoulli_setT.
End bernoulli.
Section bernoulli_measure.
Context {R : realType}.
Variables (p : R) (p0 : (0 <= p)%R) (p1 : ((NngNum p0)%:num <= 1)%R).
Lemma bernoulli_dirac : bernoulli p = measure_add
(mscale (NngNum p0) \d_true) (mscale (onem_nonneg p1) \d_false).
Proof.
apply/funext => U; rewrite /bernoulli; case: ifPn => [p01|]; last first.
by rewrite p0/= p1.
rewrite measure_addE/= /mscale/=.
have := @subsetT _ U; rewrite setT_bool => UT.
have [->|->|->|->] /= := subset_set2 UT.
- rewrite -esum_fset//=; last by move=> b; rewrite lee_fin bernoulli_pmf_ge0.
by rewrite esum_set0 2!measure0 2!mule0 adde0.
- rewrite -esum_fset//=; last by move=> b; rewrite lee_fin bernoulli_pmf_ge0.
rewrite esum_set1/= ?lee_fin// 2!diracE mem_set//= memNset//= mule0 adde0.
by rewrite mule1.
- rewrite -esum_fset//=; last by move=> b; rewrite lee_fin bernoulli_pmf_ge0.
rewrite esum_set1/= ?lee_fin ?subr_ge0// 2!diracE memNset//= mem_set//=.
by rewrite mule0 add0e mule1.
- rewrite fsbigU//=; last by move=> x [->].
by rewrite 2!fsbig_set1/= -setT_bool 2!diracT !mule1.
Qed.
by rewrite p0/= p1.
rewrite measure_addE/= /mscale/=.
have := @subsetT _ U; rewrite setT_bool => UT.
have [->|->|->|->] /= := subset_set2 UT.
- rewrite -esum_fset//=; last by move=> b; rewrite lee_fin bernoulli_pmf_ge0.
by rewrite esum_set0 2!measure0 2!mule0 adde0.
- rewrite -esum_fset//=; last by move=> b; rewrite lee_fin bernoulli_pmf_ge0.
rewrite esum_set1/= ?lee_fin// 2!diracE mem_set//= memNset//= mule0 adde0.
by rewrite mule1.
- rewrite -esum_fset//=; last by move=> b; rewrite lee_fin bernoulli_pmf_ge0.
rewrite esum_set1/= ?lee_fin ?subr_ge0// 2!diracE memNset//= mem_set//=.
by rewrite mule0 add0e mule1.
- rewrite fsbigU//=; last by move=> x [->].
by rewrite 2!fsbig_set1/= -setT_bool 2!diracT !mule1.
Qed.
End bernoulli_measure.
Arguments bernoulli {R}.
Section integral_bernoulli.
Context {R : realType}.
Variables (p : R) (p01 : (0 <= p <= 1)%R).
Local Open Scope ereal_scope.
Lemma bernoulliE A : bernoulli p A = p%:E * \d_true A + (`1-p)%:E * \d_false A.
Proof.
Lemma integral_bernoulli (f : bool -> \bar R) : (forall x, 0 <= f x) ->
\int[bernoulli p]_y (f y) = p%:E * f true + (`1-p)%:E * f false.
Proof.
move=> f0; case/andP : p01 => p0 p1; rewrite bernoulli_dirac/=.
rewrite ge0_integral_measure_sum// 2!big_ord_recl/= big_ord0 adde0/=.
by rewrite !ge0_integral_mscale//= !integral_dirac//= !diracT !mul1e.
Qed.
rewrite ge0_integral_measure_sum// 2!big_ord_recl/= big_ord0 adde0/=.
by rewrite !ge0_integral_mscale//= !integral_dirac//= !diracT !mul1e.
Qed.
End integral_bernoulli.
Section measurable_bernoulli.
Local Open Scope ring_scope.
Variable R : realType.
Implicit Type p : R.
Lemma measurable_bernoulli :
measurable_fun setT (bernoulli : R -> pprobability bool R).
Proof.
apply: (@measurability _ _ _ _ _ _
(@pset _ _ _ : set (set (pprobability _ R)))) => //.
move=> _ -[_ [r r01] [Ys mYs <-]] <-; apply: emeasurable_fun_infty_o => //=.
apply: measurable_fun_if => //=.
by apply: measurable_and => //; exact: measurable_fun_ler.
apply: (eq_measurable_fun (fun t =>
\sum_(b <- fset_set Ys) (bernoulli_pmf t b)%:E)).
move=> x /set_mem[_/= x01].
by rewrite fsbig_finite//=.
apply: emeasurable_fun_sum => n.
move=> k Ysk; apply/measurableT_comp => //.
exact: measurable_bernoulli_pmf.
Qed.
(@pset _ _ _ : set (set (pprobability _ R)))) => //.
move=> _ -[_ [r r01] [Ys mYs <-]] <-; apply: emeasurable_fun_infty_o => //=.
apply: measurable_fun_if => //=.
by apply: measurable_and => //; exact: measurable_fun_ler.
apply: (eq_measurable_fun (fun t =>
\sum_(b <- fset_set Ys) (bernoulli_pmf t b)%:E)).
move=> x /set_mem[_/= x01].
by rewrite fsbig_finite//=.
apply: emeasurable_fun_sum => n.
move=> k Ysk; apply/measurableT_comp => //.
exact: measurable_bernoulli_pmf.
Qed.
Lemma measurable_bernoulli2 U : measurable U ->
measurable_fun setT (bernoulli ^~ U : R -> \bar R).
Proof.
End measurable_bernoulli.
Arguments measurable_bernoulli {R}.
Section binomial_pmf.
Local Open Scope ring_scope.
Context {R : realType} (n : nat) (p : R).
Definition binomial_pmf k := p ^+ k * (`1-p) ^+ (n - k) *+ 'C(n, k).
Lemma binomial_pmf_ge0 k (p01 : (0 <= p <= 1)%R) : 0 <= binomial_pmf k.
Proof.
End binomial_pmf.
Lemma measurable_binomial_pmf {R : realType} D n k :
measurable_fun D (@binomial_pmf R n ^~ k).
Proof.
apply: (@measurableT_comp _ _ _ _ _ _ (fun x : R => x *+ 'C(n, k))%R) => /=.
exact: natmul_measurable.
by apply: measurable_funM => //; apply: measurable_funX; exact: measurable_funB.
Qed.
exact: natmul_measurable.
by apply: measurable_funM => //; apply: measurable_funX; exact: measurable_funB.
Qed.
Definition binomial_prob {R : realType} (n : nat) (p : R) : set nat -> \bar R :=
fun U => if (0 <= p <= 1)%R then
\esum_(k in U) (binomial_pmf n p k)%:E else \d_0%N U.
Section binomial.
Context {R : realType} (n : nat) (p : R).
Local Open Scope ereal_scope.
Local Notation binomial := (binomial_prob n p).
Let binomial0 : binomial set0 = 0.
Let binomial_ge0 U : 0 <= binomial U.
Proof.
rewrite /binomial; case: ifPn => // p01; apply: esum_ge0 => /= k Uk.
by rewrite lee_fin binomial_pmf_ge0.
Qed.
by rewrite lee_fin binomial_pmf_ge0.
Qed.
Let binomial_sigma_additive : semi_sigma_additive binomial.
Proof.
move=> F mF tF mUF; rewrite /binomial; case: ifPn => p01; last first.
exact: measure_semi_sigma_additive.
apply: cvg_toP.
apply: ereal_nondecreasing_is_cvgn => a b ab.
apply: lee_sum_nneg_natr => // k _ _.
by apply: esum_ge0 => /= ? _; exact: binomial_pmf_ge0.
by rewrite nneseries_sum_bigcup// => i; rewrite lee_fin binomial_pmf_ge0.
Qed.
exact: measure_semi_sigma_additive.
apply: cvg_toP.
apply: ereal_nondecreasing_is_cvgn => a b ab.
apply: lee_sum_nneg_natr => // k _ _.
by apply: esum_ge0 => /= ? _; exact: binomial_pmf_ge0.
by rewrite nneseries_sum_bigcup// => i; rewrite lee_fin binomial_pmf_ge0.
Qed.
HB.instance Definition _ := isMeasure.Build _ _ _ binomial
binomial0 binomial_ge0 binomial_sigma_additive.
Let binomial_setT : binomial [set: _] = 1.
Proof.
rewrite /binomial; case: ifPn; last by move=> _; rewrite probability_setT.
move=> p01; rewrite /binomial_pmf.
have pkn k : 0%R <= (p ^+ k * `1-p ^+ (n - k) *+ 'C(n, k))%:E.
case/andP : p01 => p0 p1.
by rewrite lee_fin mulrn_wge0// mulr_ge0 ?exprn_ge0 ?subr_ge0.
rewrite (esumID `I_n.+1)// [X in _ + X]esum1 ?adde0; last first.
by move=> /= k [_ /negP]; rewrite -leqNgt => nk; rewrite bin_small.
rewrite setTI esum_fset// -fsbig_ord//=.
under eq_bigr do rewrite mulrC.
rewrite sumEFin -exprDn_comm; last exact: mulrC.
by rewrite addrC add_onemK expr1n.
Qed.
move=> p01; rewrite /binomial_pmf.
have pkn k : 0%R <= (p ^+ k * `1-p ^+ (n - k) *+ 'C(n, k))%:E.
case/andP : p01 => p0 p1.
by rewrite lee_fin mulrn_wge0// mulr_ge0 ?exprn_ge0 ?subr_ge0.
rewrite (esumID `I_n.+1)// [X in _ + X]esum1 ?adde0; last first.
by move=> /= k [_ /negP]; rewrite -leqNgt => nk; rewrite bin_small.
rewrite setTI esum_fset// -fsbig_ord//=.
under eq_bigr do rewrite mulrC.
rewrite sumEFin -exprDn_comm; last exact: mulrC.
by rewrite addrC add_onemK expr1n.
Qed.
HB.instance Definition _ :=
@Measure_isProbability.Build _ _ R binomial binomial_setT.
End binomial.
Section binomial_probability.
Local Open Scope ring_scope.
Context {R : realType} (n : nat) (p : R)
(p0 : (0 <= p)%R) (p1 : ((NngNum p0)%:num <= 1)%R).
Definition bin_prob (k : nat) : {nonneg R} :=
((NngNum p0)%:num ^+ k * (NngNum (onem_ge0 p1))%:num ^+ (n - k)%N *+ 'C(n, k))%:nng.
Lemma bin_prob0 : bin_prob 0 = ((NngNum (onem_ge0 p1))%:num^+n)%:nng.
Lemma bin_prob1 : bin_prob 1 =
((NngNum p0)%:num * (NngNum (onem_ge0 p1))%:num ^+ n.-1 *+ n)%:nng.
Lemma binomial_msum :
binomial_prob n p = msum (fun k => mscale (bin_prob k) \d_k) n.+1.
Proof.
apply/funext => U.
rewrite /binomial_prob; case: ifPn => [_|]; last by rewrite p1 p0.
rewrite /msum/= /mscale/= /binomial_pmf.
have pkn k : (0%R <= (p ^+ k * `1-p ^+ (n - k) *+ 'C(n, k))%:E)%E.
by rewrite lee_fin mulrn_wge0// mulr_ge0 ?exprn_ge0 ?subr_ge0.
rewrite (esumID `I_n.+1)//= [X in _ + X]esum1 ?adde0; last first.
by move=> /= k [_ /negP]; rewrite -leqNgt => nk; rewrite bin_small.
rewrite esum_mkcondl esum_fset//; last by move=> i /= _; case: ifPn.
rewrite -fsbig_ord//=; apply: eq_bigr => i _.
by rewrite diracE; case: ifPn => /= iU; [rewrite mule1|rewrite mule0].
Qed.
rewrite /binomial_prob; case: ifPn => [_|]; last by rewrite p1 p0.
rewrite /msum/= /mscale/= /binomial_pmf.
have pkn k : (0%R <= (p ^+ k * `1-p ^+ (n - k) *+ 'C(n, k))%:E)%E.
by rewrite lee_fin mulrn_wge0// mulr_ge0 ?exprn_ge0 ?subr_ge0.
rewrite (esumID `I_n.+1)//= [X in _ + X]esum1 ?adde0; last first.
by move=> /= k [_ /negP]; rewrite -leqNgt => nk; rewrite bin_small.
rewrite esum_mkcondl esum_fset//; last by move=> i /= _; case: ifPn.
rewrite -fsbig_ord//=; apply: eq_bigr => i _.
by rewrite diracE; case: ifPn => /= iU; [rewrite mule1|rewrite mule0].
Qed.
Lemma binomial_probE U : binomial_prob n p U =
(\sum_(k < n.+1) (bin_prob k)%:num%:E * (\d_(nat_of_ord k) U))%E.
Proof.
Lemma integral_binomial (f : nat -> \bar R) : (forall x, 0 <= f x)%E ->
(\int[binomial_prob n p]_y (f y) =
\sum_(k < n.+1) (bin_prob k)%:num%:E * f k)%E.
Proof.
move=> f0; rewrite binomial_msum ge0_integral_measure_sum//=.
apply: eq_bigr => i _.
by rewrite ge0_integral_mscale//= integral_dirac//= diracT mul1e.
Qed.
apply: eq_bigr => i _.
by rewrite ge0_integral_mscale//= integral_dirac//= diracT mul1e.
Qed.
End binomial_probability.
Lemma integral_binomial_prob (R : realType) n p U : (0 <= p <= 1)%R ->
(\int[binomial_prob n p]_y \d_(0 < y)%N U =
bernoulli (1 - `1-p ^+ n) U :> \bar R)%E.
Proof.
move=> /andP[p0 p1]; rewrite bernoulliE//=; last first.
rewrite subr_ge0 exprn_ile1//=; [|exact/onem_ge0|exact/onem_le1].
by rewrite lerBlDr addrC -lerBlDr subrr; exact/exprn_ge0/onem_ge0.
rewrite (@integral_binomial _ n p _ _ (fun y => \d_(1 <= y)%N U))//.
rewrite !big_ord_recl/=.
rewrite expr0 mul1r subn0 bin0 ltnn mulr1n addrC.
rewrite onemD opprK onem1 add0r; congr +%E.
rewrite /bump; under eq_bigr do rewrite leq0n add1n ltnS leq0n.
rewrite -ge0_sume_distrl; last first.
move=> i _.
by apply/mulrn_wge0/mulr_ge0; apply/exprn_ge0 => //; exact/onem_ge0.
congr *%E.
transitivity (\sum_(i < n.+1) (`1-p ^+ (n - i) * p ^+ i *+ 'C(n, i))%:E -
(`1-p ^+ n)%:E)%E.
rewrite big_ord_recl/=.
rewrite expr0 mulr1 subn0 bin0 mulr1n addrAC -EFinD subrr add0e.
by rewrite /bump; under [RHS]eq_bigr do rewrite leq0n add1n mulrC.
rewrite sumEFin -(@exprDn_comm _ `1-p p n)//.
by rewrite subrK expr1n.
by rewrite /GRing.comm/onem mulrC.
Qed.
rewrite subr_ge0 exprn_ile1//=; [|exact/onem_ge0|exact/onem_le1].
by rewrite lerBlDr addrC -lerBlDr subrr; exact/exprn_ge0/onem_ge0.
rewrite (@integral_binomial _ n p _ _ (fun y => \d_(1 <= y)%N U))//.
rewrite !big_ord_recl/=.
rewrite expr0 mul1r subn0 bin0 ltnn mulr1n addrC.
rewrite onemD opprK onem1 add0r; congr +%E.
rewrite /bump; under eq_bigr do rewrite leq0n add1n ltnS leq0n.
rewrite -ge0_sume_distrl; last first.
move=> i _.
by apply/mulrn_wge0/mulr_ge0; apply/exprn_ge0 => //; exact/onem_ge0.
congr *%E.
transitivity (\sum_(i < n.+1) (`1-p ^+ (n - i) * p ^+ i *+ 'C(n, i))%:E -
(`1-p ^+ n)%:E)%E.
rewrite big_ord_recl/=.
rewrite expr0 mulr1 subn0 bin0 mulr1n addrAC -EFinD subrr add0e.
by rewrite /bump; under [RHS]eq_bigr do rewrite leq0n add1n mulrC.
rewrite sumEFin -(@exprDn_comm _ `1-p p n)//.
by rewrite subrK expr1n.
by rewrite /GRing.comm/onem mulrC.
Qed.
Lemma measurable_binomial_prob (R : realType) (n : nat) :
measurable_fun setT (binomial_prob n : R -> pprobability _ _).
Proof.
apply: (@measurability _ _ _ _ _ _
(@pset _ _ _ : set (set (pprobability _ R)))) => //.
move=> _ -[_ [r r01] [Ys mYs <-]] <-; apply: emeasurable_fun_infty_o => //=.
rewrite /binomial_prob/=.
set f := (X in measurable_fun _ X).
apply: measurable_fun_if => //=.
by apply: measurable_and => //; exact: measurable_fun_ler.
apply: (eq_measurable_fun (fun t =>
\sum_(k <oo | k \in Ys) (binomial_pmf n t k)%:E))%E.
move=> x /set_mem[_/= x01].
rewrite nneseries_esum// -1?[in RHS](set_mem_set Ys)// => k kYs.
by rewrite lee_fin binomial_pmf_ge0.
apply: ge0_emeasurable_fun_sum.
by move=> k x/= [_ x01] _; rewrite lee_fin binomial_pmf_ge0.
move=> k Ysk; apply/measurableT_comp => //.
exact: measurable_binomial_pmf.
Qed.
(@pset _ _ _ : set (set (pprobability _ R)))) => //.
move=> _ -[_ [r r01] [Ys mYs <-]] <-; apply: emeasurable_fun_infty_o => //=.
rewrite /binomial_prob/=.
set f := (X in measurable_fun _ X).
apply: measurable_fun_if => //=.
by apply: measurable_and => //; exact: measurable_fun_ler.
apply: (eq_measurable_fun (fun t =>
\sum_(k <oo | k \in Ys) (binomial_pmf n t k)%:E))%E.
move=> x /set_mem[_/= x01].
rewrite nneseries_esum// -1?[in RHS](set_mem_set Ys)// => k kYs.
by rewrite lee_fin binomial_pmf_ge0.
apply: ge0_emeasurable_fun_sum.
by move=> k x/= [_ x01] _; rewrite lee_fin binomial_pmf_ge0.
move=> k Ysk; apply/measurableT_comp => //.
exact: measurable_binomial_pmf.
Qed.
Section uniform_probability.
Local Open Scope ring_scope.
Context (R : realType) (a b : R).
Definition uniform_pdf x := if a <= x <= b then (b - a)^-1 else 0.
Lemma uniform_pdf_ge0 x : a < b -> 0 <= uniform_pdf x.
Proof.
Lemma measurable_uniform_pdf : measurable_fun setT uniform_pdf.
Proof.
rewrite /uniform_pdf /=; apply: measurable_fun_if => //=.
by apply: measurable_and => //; exact: measurable_fun_ler.
Qed.
by apply: measurable_and => //; exact: measurable_fun_ler.
Qed.
Local Notation mu := lebesgue_measure.
Lemma integral_uniform_pdf U :
(\int[mu]_(x in U) (uniform_pdf x)%:E =
\int[mu]_(x in U `&` `[a, b]) (uniform_pdf x)%:E)%E.
Proof.
rewrite [RHS]integral_mkcondr/=; apply: eq_integral => x xU.
rewrite patchE; case: ifPn => //.
rewrite notin_setE/= in_itv/= => /negP/negbTE xab.
by rewrite /uniform_pdf xab.
Qed.
rewrite patchE; case: ifPn => //.
rewrite notin_setE/= in_itv/= => /negP/negbTE xab.
by rewrite /uniform_pdf xab.
Qed.
Lemma integral_uniform_pdf1 A (ab : a < b) : `[a, b] `<=` A ->
(\int[mu]_(x in A) (uniform_pdf x)%:E = 1)%E.
Proof.
move=> abA; rewrite integral_uniform_pdf setIidr//.
rewrite (eq_integral (fun=> (b - a)^-1%:E)); last first.
by move=> x; rewrite inE/= in_itv/= /uniform_pdf => ->.
rewrite integral_cst//= lebesgue_measure_itv/= lte_fin.
by rewrite ab -EFinD -EFinM mulVf// gt_eqF// subr_gt0.
Qed.
rewrite (eq_integral (fun=> (b - a)^-1%:E)); last first.
by move=> x; rewrite inE/= in_itv/= /uniform_pdf => ->.
rewrite integral_cst//= lebesgue_measure_itv/= lte_fin.
by rewrite ab -EFinD -EFinM mulVf// gt_eqF// subr_gt0.
Qed.
Definition uniform_prob (ab : a < b) : set _ -> \bar R :=
fun U => (\int[mu]_(x in U) (uniform_pdf x)%:E)%E.
Hypothesis ab : (a < b)%R.
Let uniform0 : uniform_prob ab set0 = 0.
Proof.
Let uniform_ge0 U : (0 <= uniform_prob ab U)%E.
Proof.
Lemma integrable_uniform_pdf :
mu.-integrable setT (fun x => (uniform_pdf x)%:E).
Proof.
apply/integrableP; split.
by apply: measurableT_comp => //; exact: measurable_uniform_pdf.
under eq_integral.
move=> x _; rewrite gee0_abs//; last by rewrite lee_fin uniform_pdf_ge0.
over.
by rewrite /= integral_uniform_pdf1 ?ltry// -subr_gt0.
Qed.
by apply: measurableT_comp => //; exact: measurable_uniform_pdf.
under eq_integral.
move=> x _; rewrite gee0_abs//; last by rewrite lee_fin uniform_pdf_ge0.
over.
by rewrite /= integral_uniform_pdf1 ?ltry// -subr_gt0.
Qed.
Let uniform_sigma_additive : semi_sigma_additive (uniform_prob ab).
Proof.
move=> /= F mF tF mUF; rewrite /uniform_prob; apply: cvg_toP.
apply: ereal_nondecreasing_is_cvgn => m n mn.
apply: lee_sum_nneg_natr => // k _ _.
by apply: integral_ge0 => /= x Fkx; rewrite lee_fin uniform_pdf_ge0.
rewrite ge0_integral_bigcup//=.
- apply: measurable_funTS; apply: measurableT_comp => //.
exact: measurable_uniform_pdf.
- by move=> x _; rewrite lee_fin uniform_pdf_ge0.
Qed.
apply: ereal_nondecreasing_is_cvgn => m n mn.
apply: lee_sum_nneg_natr => // k _ _.
by apply: integral_ge0 => /= x Fkx; rewrite lee_fin uniform_pdf_ge0.
rewrite ge0_integral_bigcup//=.
- apply: measurable_funTS; apply: measurableT_comp => //.
exact: measurable_uniform_pdf.
- by move=> x _; rewrite lee_fin uniform_pdf_ge0.
Qed.
HB.instance Definition _ := isMeasure.Build _ _ _ (uniform_prob ab)
uniform0 uniform_ge0 uniform_sigma_additive.
Let uniform_setT : uniform_prob ab [set: _] = 1%:E.
Proof.
HB.instance Definition _ := @Measure_isProbability.Build _ _ R
(uniform_prob ab) uniform_setT.
Lemma dominates_uniform_prob : uniform_prob ab `<< mu.
Proof.
move=> A mA muA0; rewrite /uniform_prob integral_uniform_pdf.
apply/eqP; rewrite eq_le; apply/andP; split; last first.
apply: integral_ge0 => x [Ax /=]; rewrite in_itv /= => xab.
by rewrite lee_fin uniform_pdf_ge0.
apply: (@le_trans _ _
(\int[mu]_(x in A `&` `[a, b]%classic) (b - a)^-1%:E))%E; last first.
rewrite integral_cst//= ?mul1e//.
by rewrite pmule_rle0 ?lte_fin ?invr_gt0// ?subr_gt0// -muA0 measureIl.
exact: measurableI.
apply: ge0_le_integral => //=.
- exact: measurableI.
- by move=> x [Ax]; rewrite /= in_itv/= => axb; rewrite lee_fin uniform_pdf_ge0.
- by apply/EFin_measurable_fun/measurable_funTS; exact: measurable_uniform_pdf.
- by move=> x [Ax _]; rewrite lee_fin invr_ge0// ltW// subr_gt0.
- by move=> x [Ax]; rewrite in_itv/= /uniform_pdf => ->.
Qed.
apply/eqP; rewrite eq_le; apply/andP; split; last first.
apply: integral_ge0 => x [Ax /=]; rewrite in_itv /= => xab.
by rewrite lee_fin uniform_pdf_ge0.
apply: (@le_trans _ _
(\int[mu]_(x in A `&` `[a, b]%classic) (b - a)^-1%:E))%E; last first.
rewrite integral_cst//= ?mul1e//.
by rewrite pmule_rle0 ?lte_fin ?invr_gt0// ?subr_gt0// -muA0 measureIl.
exact: measurableI.
apply: ge0_le_integral => //=.
- exact: measurableI.
- by move=> x [Ax]; rewrite /= in_itv/= => axb; rewrite lee_fin uniform_pdf_ge0.
- by apply/EFin_measurable_fun/measurable_funTS; exact: measurable_uniform_pdf.
- by move=> x [Ax _]; rewrite lee_fin invr_ge0// ltW// subr_gt0.
- by move=> x [Ax]; rewrite in_itv/= /uniform_pdf => ->.
Qed.
Let integral_uniform_indic E : measurable E ->
(\int[uniform_prob ab]_x (\1_E x)%:E =
(b - a)^-1%:E * \int[mu]_(x in `[a, b]) (\1_E x)%:E)%E.
Proof.
move=> mE; rewrite integral_indic//= /uniform_prob setIT -ge0_integralZl//=.
- rewrite [LHS]integral_mkcond/= [RHS]integral_mkcond/=.
apply: eq_integral => x _; rewrite !patchE; case: ifPn => xE.
case: ifPn.
rewrite inE/= in_itv/= => xab.
by rewrite /uniform_pdf xab indicE xE mule1.
by rewrite notin_setE/= in_itv/= => /negP/negbTE; rewrite /uniform_pdf => ->.
case: ifPn => //.
by rewrite inE/= in_itv/= => axb; rewrite indicE (negbTE xE) mule0.
- exact/EFin_measurable_fun/measurable_indic.
- by move=> x _; rewrite lee_fin.
- by rewrite lee_fin invr_ge0// ltW// subr_gt0.
Qed.
- rewrite [LHS]integral_mkcond/= [RHS]integral_mkcond/=.
apply: eq_integral => x _; rewrite !patchE; case: ifPn => xE.
case: ifPn.
rewrite inE/= in_itv/= => xab.
by rewrite /uniform_pdf xab indicE xE mule1.
by rewrite notin_setE/= in_itv/= => /negP/negbTE; rewrite /uniform_pdf => ->.
case: ifPn => //.
by rewrite inE/= in_itv/= => axb; rewrite indicE (negbTE xE) mule0.
- exact/EFin_measurable_fun/measurable_indic.
- by move=> x _; rewrite lee_fin.
- by rewrite lee_fin invr_ge0// ltW// subr_gt0.
Qed.
Let integral_uniform_nnsfun (f : {nnsfun _ >-> R}) :
(\int[uniform_prob ab]_x (f x)%:E =
(b - a)^-1%:E * \int[mu]_(x in `[a, b]) (f x)%:E)%E.
Proof.
under [LHS]eq_integral do rewrite fimfunE -fsumEFin//.
rewrite [LHS]ge0_integral_fsum//; last 2 first.
- by move=> r; exact/EFin_measurable_fun/measurableT_comp.
- by move=> n x _; rewrite EFinM nnfun_muleindic_ge0.
rewrite -[RHS]ge0_integralZl//; last 3 first.
- exact/EFin_measurable_fun/measurable_funTS.
- by move=> x _; rewrite lee_fin.
- by rewrite lee_fin invr_ge0// ltW// subr_gt0.
under [RHS]eq_integral.
move=> x xD; rewrite fimfunE -fsumEFin// ge0_mule_fsumr; last first.
by move=> r; rewrite EFinM nnfun_muleindic_ge0.
over.
rewrite [RHS]ge0_integral_fsum//; last 2 first.
- by move=> r; apply/EFin_measurable_fun; do 2 apply/measurableT_comp => //.
- move=> n x _; rewrite EFinM mule_ge0//; last by rewrite nnfun_muleindic_ge0.
by rewrite lee_fin invr_ge0// ltW// subr_gt0.
apply: eq_fsbigr => r _; rewrite ge0_integralZl//.
- by rewrite !integralZl_indic_nnsfun//= integral_uniform_indic// muleCA.
- exact/EFin_measurable_fun/measurableT_comp.
- by move=> t _; rewrite nnfun_muleindic_ge0.
- by rewrite lee_fin invr_ge0// ltW// subr_gt0.
Qed.
rewrite [LHS]ge0_integral_fsum//; last 2 first.
- by move=> r; exact/EFin_measurable_fun/measurableT_comp.
- by move=> n x _; rewrite EFinM nnfun_muleindic_ge0.
rewrite -[RHS]ge0_integralZl//; last 3 first.
- exact/EFin_measurable_fun/measurable_funTS.
- by move=> x _; rewrite lee_fin.
- by rewrite lee_fin invr_ge0// ltW// subr_gt0.
under [RHS]eq_integral.
move=> x xD; rewrite fimfunE -fsumEFin// ge0_mule_fsumr; last first.
by move=> r; rewrite EFinM nnfun_muleindic_ge0.
over.
rewrite [RHS]ge0_integral_fsum//; last 2 first.
- by move=> r; apply/EFin_measurable_fun; do 2 apply/measurableT_comp => //.
- move=> n x _; rewrite EFinM mule_ge0//; last by rewrite nnfun_muleindic_ge0.
by rewrite lee_fin invr_ge0// ltW// subr_gt0.
apply: eq_fsbigr => r _; rewrite ge0_integralZl//.
- by rewrite !integralZl_indic_nnsfun//= integral_uniform_indic// muleCA.
- exact/EFin_measurable_fun/measurableT_comp.
- by move=> t _; rewrite nnfun_muleindic_ge0.
- by rewrite lee_fin invr_ge0// ltW// subr_gt0.
Qed.
Lemma integral_uniform (f : _ -> \bar R) :
measurable_fun setT f -> (forall x, 0 <= f x)%E ->
(\int[uniform_prob ab]_x f x = (b - a)^-1%:E * \int[mu]_(x in `[a, b]) f x)%E.
Proof.
move=> mf f0.
have [f_ [ndf_ f_f]] := approximation measurableT mf (fun y _ => f0 y).
transitivity (lim (\int[uniform_prob ab]_x (f_ n x)%:E @[n --> \oo])%E).
rewrite -monotone_convergence//=.
- apply: eq_integral => ? /[!inE] xD; apply/esym/cvg_lim => //=.
exact: f_f.
- by move=> n; exact/EFin_measurable_fun/measurable_funTS.
- by move=> n ? _; rewrite lee_fin.
- by move=> ? _ ? ? mn; rewrite lee_fin; exact/lefP/ndf_.
rewrite [X in _ = (_ * X)%E](_ : _ = lim
(\int[mu]_(x in `[a, b]) (f_ n x)%:E @[n --> \oo])%E); last first.
rewrite -monotone_convergence//=.
- by apply: eq_integral => ? /[!inE] xD; apply/esym/cvg_lim => //; exact: f_f.
- by move=> n; exact/EFin_measurable_fun/measurable_funTS.
- by move=> n ? _; rewrite lee_fin.
- by move=> ? _ ? ? /ndf_ /lefP; rewrite lee_fin.
rewrite -limeMl//.
by apply: congr_lim; apply/funext => n /=; exact: integral_uniform_nnsfun.
apply/ereal_nondecreasing_is_cvgn => x y xy; apply: ge0_le_integral => //=.
- by move=> ? _; rewrite lee_fin.
- exact/EFin_measurable_fun/measurable_funTS.
- by move=> ? _; rewrite lee_fin.
- exact/EFin_measurable_fun/measurable_funTS.
- by move=> ? _; rewrite lee_fin; move/ndf_ : xy => /lefP.
Qed.
have [f_ [ndf_ f_f]] := approximation measurableT mf (fun y _ => f0 y).
transitivity (lim (\int[uniform_prob ab]_x (f_ n x)%:E @[n --> \oo])%E).
rewrite -monotone_convergence//=.
- apply: eq_integral => ? /[!inE] xD; apply/esym/cvg_lim => //=.
exact: f_f.
- by move=> n; exact/EFin_measurable_fun/measurable_funTS.
- by move=> n ? _; rewrite lee_fin.
- by move=> ? _ ? ? mn; rewrite lee_fin; exact/lefP/ndf_.
rewrite [X in _ = (_ * X)%E](_ : _ = lim
(\int[mu]_(x in `[a, b]) (f_ n x)%:E @[n --> \oo])%E); last first.
rewrite -monotone_convergence//=.
- by apply: eq_integral => ? /[!inE] xD; apply/esym/cvg_lim => //; exact: f_f.
- by move=> n; exact/EFin_measurable_fun/measurable_funTS.
- by move=> n ? _; rewrite lee_fin.
- by move=> ? _ ? ? /ndf_ /lefP; rewrite lee_fin.
rewrite -limeMl//.
by apply: congr_lim; apply/funext => n /=; exact: integral_uniform_nnsfun.
apply/ereal_nondecreasing_is_cvgn => x y xy; apply: ge0_le_integral => //=.
- by move=> ? _; rewrite lee_fin.
- exact/EFin_measurable_fun/measurable_funTS.
- by move=> ? _; rewrite lee_fin.
- exact/EFin_measurable_fun/measurable_funTS.
- by move=> ? _; rewrite lee_fin; move/ndf_ : xy => /lefP.
Qed.
End uniform_probability.